WHAT IS A RATE?

Concepts

• A rate is a vector if it has a direction
  

• Vectors have both magnitude and direction
  
• Vectors can be mathematically manipulated
  
• Vectors can be used to describe properties of motion

Materials

Required skills

Background

Everyday we use rates to describe events that happen over a period of time. Perhaps you drove 65 miles per hour to school this morning because you were running late, or you rode on an amusement park ride last weekend that went 80 miles per hour. Each time you describe a change in position or velocity with respect to time you have used a rate to do it. Rates can be described mathematically and graphically. A formula such as: velocity = change in position / change in time, gives a mathematical description of a rate. Graphically, rates are represented by arrows, whose length equals the magnitude of the rate and whose orientation on an axis represents the direction in which the object at that rate is moving. These arrows are called vectors. Vectors have both magnitude and direction, and therefore describe rates, such as velocity and acceleration. If a property can be described graphically but does not have a direction it is called a scalar. Scalars have magnitude but no direction, such as time, mass, and temperature. So why do people use vectors instead of just mathematical expressions? Actually people use both. A perfect example of this is the velocity maps used to describe how fast and in what direction GPS sites on the Earth's plates are moving. [link to velocity map] Scientists first calculate velocities for each GPS site in centimeters per year, using mathematical equations. Then, they place the velocities, in the form of vectors, on a map of the Earth. This allows them to get a better mental picture of what is happening to the plates world wide. These velocity maps can be made for any area where the displacement of a set point over a period of time is known. A displacement is an overall change in position from the start of a measurement to the end of a measurement. In the case of a velocity map, the measurement is a length of time, most commonly a year.

Vectors, which are rates, can be added, subtracted, and multiplied. There are a few rules when using vectors: 1) when adding two vectors they must be placed head to tail; 2) when subtracting two vectors, change the sign of one vector and then add it to the other; and 3) if you multiply a vector by a scalar, only the magnitude is changed, unless the scalar has a negative value and then the direction of the vector is reversed as well.

These concepts may seem new to you, but you have probably already used them without knowing it. Have you ever found one length of a triangle by knowing the values of its other two sides? Then you have used vectors. The key to successfully using vectors is in understanding that you are basically solving for the third side of a triangle. The third side, in this case, is called the displacement vector. Remember, a vector has a magnitude (length) and direction; and a rate is a vector, so it also has a magnitude and direction.

Helpful Formulas

Procedure

Answer the following problems. Be sure to draw out the vectors on the graph paper for each problem that applies.

Theory Questions:

2. Another way to add vectors is to place them tail to tail and then form a parallelogram. The sum of the vectors is the diagonal of the parallelogram, or the resultant motion. Why is this method equivalent to the head-to-tail method? Apply this to the following problem and solve for the diagonal.

Application Questions:

2. A GPS station moves north an average of 15 mm in 3 years. It also moves east an average of 36 mm in the same amount of time. What is the station's displacement? What is its overall average velocity per year?

Challenge:

Questions to Answer

Last modified on 8/13/98 by Maggi Glasscoe (scignedu@jpl.nasa.gov)